Finite-sided Dirichlet domains and Anosov representations

In his study of uniform lattices in SL(n,R), Selberg described a construction of a polyhedral fundamental domain for any discrete subgroup of SL(n,R) in the projective model for the associated symmetric space. This construction is closely related to the construction of Dirichlet domains. However, for non-uniform lattices, these polyhedral domains can have infinitely many sides.

As Dirichlet domains are always finite-sided for convex-cocompact subgroups in rank one Lie groups, it is natural to ask if the Dirichlet-Selberg domains are finite-sided for Anosov subgroups. Very simple Anosov subgroups in SL(3,R) can have infinite-sided Dirichlet-Selberg domains. We introduce a condition that in general is stronger than the Anosov property that implies that all Dirichlet-Selberg domains are finite sided in a strong sense. This condition turns out to be equivalent to some Anosov property if the subgroup lies in Sp(2n,R) or O(n,n). The proof of this result can be generalized to show that certain Dirichlet domains for Finsler metrics are finite-sided for certain Anosov subgroups. This is a joint work with Max Riestenberg.